PRINCIPIA RECURSIONIS (v5.0) A Typed, Variational, and Testable Operator Formalism for Agentic State Evolution: Valuation Expansion, Desire, Drive, Meaning Relief, and Root Regulation

PRINCIPIA RECURSIONIS (v5.0)

A Typed, Variational, and Testable Operator Formalism for Agentic State Evolution: Valuation Expansion, Desire, Drive, Meaning Relief, and Root Regulation

Author: Gabriel Acosta

Affiliation: Aurimens Research Group, Toronto, Ontario, Canada

Date: Oct, 2026

Correspondence: aurimensgroup@gmail.com

Version: v5.0

Status: Preprint draft for academic circulation

Keywords: recursive dynamics, information geometry, active inference, motivation modeling, control theory, cognitive systems, agentic recursion, entropy minimization, meaning theory, self-regulation, integrated information, free energy principle

NOTE TO READERS: TWO REGISTERS, ONE PAPER

This paper is written in two simultaneous registers. The technical register addresses physicists, mathematicians, cognitive scientists, and computational neuroscientists; it contains formal definitions, typed operators, proofs, and falsifiable predictions. The accessible register appears in shaded Plain Language Summary blocks at the end of each major section; it translates the core ideas into everyday language for educated non-specialists, policymakers, practitioners, and curious general readers.

Both registers carry the same intellectual content. Neither is subordinate. Readers are encouraged to read whichever serves their purposes, or both.

ABSTRACT

This work presents Principia Recursionis, a typed, variational, and empirically testable operator formalism describing how any resource-bounded, adaptive system—ranging from a single neuron to a human agent—evolves its internal state across time. The central claim is this: any system subject to simultaneous bounds on information processing, thermodynamic work, and model complexity must, when operating under constrained optimization, generate an update rule of the form described here. That update rule—the master recursion—is not postulated but derived.

What is proved, and what is interpreted: The variational derivation yields abstract Lagrange multipliers and dual co-state variables. The mathematical structure of these variables—their roles as gain operators, salience fields, kinetic scalars, and proximal maps—is derived from first principles. The naming of these structures as Valuation Expansion (A_L), Desire (Z), Drive (J), Meaning (M), and Root Regulation (R_S) is an interpretive act, grounded in functional correspondence with established neuroscientific constructs and empirical measurability, but not logically forced by the mathematics alone. This distinction is maintained throughout the paper.

The formalism supports five rigorously typed operators, each with measurable neural and behavioral proxies: (i) Valuation Expansion (A_L), a diagonal gain operator extending utility weights from self-centered to other- and system-inclusive; (ii) Meaning (M), an entropy-relief functional quantifying compression gain, empowerment, and goal alignment; (iii) Drive (J), a bounded kinetic capacity modulated by perceived progress independent of immediate reward; (iv) Desire (Z), a motivational salience field encoding cue-triggered wanting distinct from deliberative value; and (v) Root Regulation (R_S), a homeostatic precision stabilizer linking physiological coherence to computational efficiency.

The paper demonstrates that the recursion is consistent with—and reduces to—unitary quantum dynamics, classical Hamiltonian flow, Friston’s free energy minimization (Friston, 2010), entropy-regularized reinforcement learning (Haarnoja et al., 2018), and replicator dynamics (Taylor & Jonker, 1978) as special limiting cases. These are framed honestly as consistency demonstrations, not proofs that the recursion uniquely generates those theories. The conditions under which those reductions hold are fully specified, and all reductions require explicit operator choices that are motivated but not uniquely determined by the variational structure.

The key empirical prediction—the Drive–Progress Coupling Law—asserts that an agent’s vigor and persistence increase with perceived structural learning progress, independent of reward rate and metabolic cost. This prediction is novel, directly falsifiable, and cannot be explained by existing reward-rate vigor theories (Niv et al., 2007) or opportunity-cost models (Shadmehr et al., 2016). A pre-registerable experimental protocol is fully specified.

Mathematical properties established include: firm non-expansiveness of the proximal block (Parikh & Boyd, 2014); bounded forward invariance under explicit gain caps and drive ceiling; explicit contraction conditions with concrete parameter bounds for the linear-Gaussian instantiation; and sufficient conditions for quasiperiodic (torus) attractors under near-integrable propagation.

ACADEMIC CLASSIFICATION CODES

Primary: 94A17 (Measures of Information, Entropy); 93A15 (Large-Scale Systems); 68T07 (Artificial Neural Networks and Deep Learning)

Secondary: 37N40 (Dynamical Systems in Optimization and Control); 92C20 (Neural Biophysics); 03G10 (Logical Foundations of Algebraic Systems)

PROLEGOMENA: WHY PROCESS, NOT SUBSTANCE

1. The Historical Turn from Substance to Process

Western metaphysics has long oscillated between two poles. Substance ontology—the view that reality is composed of enduring things with intrinsic properties—dominated from Aristotle through Descartes to Newton. In this framework, the fundamental units are objects that persist through time and move through a fixed background. The cognitive revolution of the twentieth century inherited this structure: mental states were conceived as internal representations, fixed snapshots of a self that persists.

Process ontology, advanced by Whitehead (1929) in Process and Reality, offers a rival account: what ultimately exists is not things but events—acts of becoming, not states of being. Each entity is constituted by its relations to other entities and to its own prior states, not by any intrinsic essence it possesses independently. This view was independently discovered by the cyberneticians of the mid-twentieth century: Wiener (1948), Ashby (1956), and von Foerster all argued that what persists in living and cognitive systems is not matter but organization—recursive patterns of self-maintenance that counter local entropy (Maturana & Varela, 1980).

The modern picture in physics converges with this view. Fields have replaced particles; probabilities have replaced certainties; information has replaced matter as the fundamental currency of physical law (Wheeler, 1990; Rovelli, 1996). The discovery that entropy and information obey the same formal structure (Jaynes, 1957) suggests that the universe’s dynamics can be read as an information-processing recursion as much as a mechanical evolution.

Principia Recursionis adopts this stance: the universe, and all adaptive agents within it, are best understood as update rules—recursive processes that map current states onto next states under constraints. Being is the name we give to the persistence of an update rule. Identity is pattern-maintenance through change.

2. The Epistemic Necessity of Recursion

Recursion is not an optional mathematical convenience. It is the structural precondition of any coherent temporal evolution. To define continuity across time—for any system, physical or cognitive—is to define a rule F such that:

t+1=Ft,t,∇t,C

where Ψ is the system’s state, F is its update law, and C represents contextual constraints. Every law of motion in physics, every learning algorithm, every biological regulatory mechanism, presupposes such a rule. The recursive operator is not invented; it is found in the structure of temporal coherence itself.

This epistemic inevitability motivates the central project: to specify the typed content of F for bounded adaptive agents, deriving which operators necessarily appear and what functional roles they must play.

3. Process as Variational Law

The bridge from process ontology to rigorous physics passes through variational principles. Nature does not compute symbolic rules; it extremizes quantities. Every known physical law can be derived from an action principle (Landau & Lifshitz, 1976). Principia Recursionis extends this insight: adaptive state evolution is the proximal solution to a composite optimization problem, constrained by bounded information, energy, and complexity.

This is the formal content of what Whitehead called the “creative advance into novelty.” The variational recursion selects, at each step, the update that minimizes free-energy-like cost while remaining within resource budgets. Reality, in this reading, is a self-optimizing recursion—not by design, but by the structure of constraint satisfaction under bounded resources.

4. The Philosophical Necessity of a Typed Formalism

Prior unified frameworks—Friston’s (2010) Free Energy Principle, Maturana and Varela’s (1980) autopoiesis, and classical cybernetics—have been criticized for lacking sufficient precision to generate falsifiable predictions (Colombo & Series, 2012; Klein, 2018). The Free Energy Principle, while mathematically rigorous at the scalar level, does not explicitly type its operators or specify how different cognitive functions (attention, desire, meaning) map onto distinct components of the functional.

Principia Recursionis addresses this by introducing explicit operator typing: every term in the master recursion has a specified domain, codomain, mathematical form, and empirical proxy. This is not merely notational discipline; it is what enables falsification. A typed operator that fails to have a measurable instantiation is a failed operator, not a gap to be filled by interpretation.

Plain Language Summary — Prolegomena

Imagine that reality is not made of things but of updates—not a world of static objects, but a world of processes that continuously rewrite themselves. A rock is not just a rock sitting there; it is a slow pattern of atomic interactions maintaining a local order. A mind is not a container of fixed thoughts; it is a living recursion of perception, prediction, error, and correction. Even a particle propagating through space is, at its deepest level, a probability amplitude updating itself according to the Schrödinger equation. Principia Recursionis takes this insight—that change is primary, not substance—and asks: if everything is an update rule, what does that rule actually look like for an agent that cares, learns, and acts? That is the question this paper formally answers.

SECTION 1: STATE SPACE, TYPES, AND NOTATION

1.1 Agentic State Definition

Let the agentic state of a recursive system at time t be:

t=t, xtS := X

where:

• =Rn represents parametric beliefs or internal model weights of a generative process px, encoding priors and predictive expectations;

• xX=Rnx represents task-relevant observables or latent semantic states interacting with the environment.

• 
  

The joint space S=X≅Rn (where n=n+nx) is a finite-dimensional real Hilbert space with inner product ⟨⋅,⋅⟩ and induced norm ∥⋅∥. All operators defined below act on S or subspaces thereof. The finite-dimensional setting ensures that all operator compositions are algebraically well-typed without requiring separate Hilbert space embeddings—a point made explicit in Subsection 1.4.

1.2 Operator Spaces and Metrics

We denote:

• LS: the space of bounded linear operators on S;

• ∥⋅∥: the Euclidean (L₂) norm on vector spaces;

• H: Shannon entropy;

• DKL⋅∥⋅: Kullback–Leibler divergence;

• proxβfv := argminy12y−v2+βfy: the proximal map of a convex functional f with penalty >0.

1.3 The Canonical Master Equation

The single, canonical form of the master recursion—used consistently throughout this paper—is:

t+=H​AL​​ΨCWJE−LMD

Notation for each component:

• H:SS is the lawful propagator—the physical or dynamical evolution kernel;

• ALLS is the valuation expansion operator—a diagonal gain matrix;

• CLS is the attention operator—a precision-weighted projection;

• ​ΨS is the gradient of the model-error loss L with respect to ;

• WJ=JWS is the drive-scaled intention vector—a policy proposal;

•  denotes precision-weighted gradient binding (defined in §3.1);

•  denotes intention injection (defined in §3.5);

•  denotes the proximal shrinkage map v↦proxβEv+LMv (defined in §3.6);

• E:SR0 is the entropy-cost functional;

• M:SR0 is the meaning functional;

• L0 is the meaning-relief coupling scalar, modulated by RS;

• D∈{0,1} is the decision gate;

• RS does not appear directly in the master equation. It acts as a meta-regulator, setting the values of  (attention temperature), L (meaning coupling), and D (decision threshold) via the equations in §3.7. This eliminates the syntactic duplication present in earlier versions and makes the operator algebra unambiguous.

1.4 Type-Safety of the Composition

A previous criticism of this formalism was that the operator types were declared but the composition was not verified to be well-typed. We address this directly.

Since S≅Rn is a single finite-dimensional Hilbert space, and since XS by construction (the task-state components are embedded in the full state), every operator acts within S. Specifically:

• ​ΨS: the gradient is a cotangent vector, identified with a tangent vector via the Riesz isomorphism on the finite-dimensional Euclidean space.

• C:SS (positive semidefinite, C21): the attention-weighted gradient ​ΨC=C​ΨS is a well-typed element of S.

• WJXS: the intention vector lives in the task-state subspace, which is a subspace of S. The injection  adds WJ to the gradient term within S, using a bounded linear mixing map MLSS,S with M2LM<∞ (see §3.5).

• ALLS: acts on the result of ⋅⊕⋅, which lives in S. Type: SS. ✓

• E−LM:SR0: the proximal map acts on the vector argument. proxβE:SS, and the forward step v↦v+LMv maps SS. Type: SS. ✓

• H:SS. ✓

• D∈{0,1} acts as a gate: the composed update is +=+DΔΨ. ✓

The full composition is well-typed. No free mixing-map choices are required.

1.5 Typed Operator Table

Operator	Type	Functional Role	Neural Analogue
Attention C	SLS	PSD precision-weighted projection	Cortical gain control; Bayesian precision
Valuation AL	SLS	Diagonal utility gain matrix	vmPFC/OFC value integration
Desire Z	SX	Motivational salience field	Mesolimbic dopamine; incentive salience
Drive J	SR00,Jmax	Bounded kinetic capacity	Striatal vigor; metabolic throughput
Intention WJ	SX	Drive-scaled policy proposal	HJB gradient / premotor plan
Entropy E	SR0	Aggregate cost functional	Helmholtz free energy; metabolic load
Meaning M	SR0	Compression + empowerment + alignment	Predictive coding efficiency
Root Regulation RS	scalar → {,L,D}	Modulates attention temp., meaning coupling, decision threshold	Allostatic homeostasis; LC-NE system
Propagator H	SS	Physical/dynamical evolution kernel	Hamiltonian / Fokker–Planck flow
Decision Gate D	S→{0,1}	Binary EVC activation	ACC gate; cortical–striatal threshold

1.6 The Interpretive Mapping: What the Math Gives vs. What Is Named

This subsection is essential and was missing from all prior versions.

The variational derivation of Section 2 yields abstract mathematical objects: Lagrange multipliers 1,2,3; co-state vectors conjugate to the three constraint functionals; a proximal map; a decision threshold. The names we assign to these objects—Valuation, Desire, Drive, Meaning, Root Regulation—are interpretive, not logically entailed by the mathematics.

This interpretive mapping is motivated by three criteria:

Criterion 1: Functional correspondence. Each named operator is matched to a neuroscientific construct with established empirical support. Valuation corresponds to the vmPFC utility-integration role (Bartra et al., 2013); Desire corresponds to the wanting/liking dissociation in mesolimbic dopamine circuits (Berridge & Robinson, 1998); Drive corresponds to the vigor framework of striatal dopamine tied to effort deployment (Niv et al., 2007).

Criterion 2: Empirical distinguishability. The constructs are operationally distinct: desire is cue-triggered, valuation is deliberative, drive is kinetic. These distinctions are measurable (see §7 for the Fisher Information analysis establishing block-diagonal identifiability).

Criterion 3: Formal orthogonality. The operators are structurally distinct in the algebra: AL is a diagonal gain matrix, Z is a salience vector field, J is a bounded scalar. Distinct mathematical types support distinct empirical roles.

No claim is made that the mathematics uniquely forces these psychological interpretations. Alternative interpretive mappings could be constructed, and testing which mapping best organizes empirical data is itself a scientific question. The framework would not be falsified by a finding that a different interpretive scheme is superior; it would be falsified by the failure of the operators’ behavioral and neural predictions (see §7.9).

Plain Language Summary — Section 1

Think of an agent—you, an animal, an AI—as existing in a “state” that captures both what it knows or believes (θ, internal model) and what’s happening around it (x, the world). The master equation is a single rule that says: given where you are now, here is how you take one step forward. That step is shaped by what you’re paying attention to (C), what you value (A_L), how much drive or energy you’re deploying (J), how meaningful the task is (M), and a gate that decides whether to act at all (D). The Root Regulation operator (R_S) is like your physiological background—sleep, nutrition, stress—which quietly adjusts the sensitivity of all the other operators without appearing explicitly in the equation.

The key transparency here is this: the math gives you abstract levers. The names we give those levers—“Valuation,” “Drive,” “Meaning”—are our best scientific interpretation. They’re grounded in decades of neuroscience, but they’re named by us, not dictated by the calculus.

SECTION 2: VARIATIONAL DERIVATION

2.1 What Is Being Derived and What Is Being Interpreted

The master recursion is derived from a resource-constrained variational principle. At each time step, the system selects an update ΔΨ and control u to solve:

ΔΨ, u EpL+EE,u−⟨AL EU, ΔΨ⟩−Z⟨Z, ΔΨ⟩

subject to:

ΔΨ=,u;+, ξN0,

and a drive-limited budget:

u∥≤JJ, 0JJmax

where AL and Z appear as modeling choices reflecting the multi-objective and salience-modulated structure of the utility function (see §2.3 for justification). They are not uniquely derived from the constraints below, but they are uniquely constrained in their functional form once the utility structure is specified.

2.2 The Three Resource Constraints

The objective (1) is the Lagrangian dual of a constrained maximization problem. The constraints, and their justification, are:

Constraint I — Bounded Information Processing:

EbitsBI

This formalizes rational inattention (Sims, 2003): cognitive systems process information at finite rates. In rate–distortion theory, exceeding BI is physically impossible; in the Lagrangian dual, the constraint introduces a KL-divergence penalty 1DKLpp0 (Ortega & Braun, 2013). This penalty is the mathematical origin of the proximal  operator.

Constraint II — Bounded Thermodynamic Work:

EWorkuBW

Biological agents have finite metabolic budgets; artificial agents have finite computational budgets. The Lagrange multiplier 2 is conjugate to thermodynamic work and introduces the convex cost 2Wu (Still et al., 2012). This constraint is what forces the drive bound JJmax in (3).

Constraint III — Bounded Model Complexity:

EKBK

Minimum description length (MDL) bounds on the agent’s internal model (Schmidhuber, 2010) yield a complexity penalty 3K (Hinton & van Camp, 1993). This constraint motivates the compression component of the Meaning functional M.

2.3 The Lagrangian Dual and Operator Emergence

Maximizing expected utility EU,u subject to the three constraints yields the Lagrangian dual:

J=EU−1Ebits−2EWork−3EMDL

Switching to minimization of the corresponding free-energy functional and applying gradient descent under the Fisher information metric (Amari, 1998) yields:

*=proxβE′​​ΨCWJ

where E′ := E−LM incorporates meaning relief, and WJ is the policy proposal scaled by Drive (see §3.5). Propagating through H and applying gate D yields the canonical master recursion.

What emerges from the math vs. what is chosen:

• The existence of a proximal shrinkage step is derived (from Constraint I).

• The form of the proximal map—proxβE—is derived (from convexity of E).

• That the gain operator on the gradient should be diagonal and represent utility weights is a modeling choice: it is the most general linear Pareto aggregator (Lotov & Miettinen, 2008), but other forms are conceivable.

• That the bias field on proposals should be interpreted as “Desire” (cue-triggered salience distinct from deliberative value) reflects a motivated interpretive choice, grounded in the neuroscience of incentive salience (Berridge & Robinson, 1998).

2.4 Why Valuation, Desire, and Intention Are Formally Orthogonal

Construct	Mathematical Role	Neuroscientific Grounding	Distinct From
AL (Valuation)	Gain on gradient features; SLS	vmPFC deliberative appraisal	Not wanting, not acting
Z (Desire)	Salience bias field; SX	Mesolimbic incentive salience	Not valuing, not planning
WJ (Intention)	Policy-optimal action direction; SX	Premotor/frontoparietal planning	Not wanting, not valuing

This orthogonality supports parameter identifiability (§7) and reflects the empirically supported dissociation between wanting, valuing, and planning documented in Berridge and Kringelbach (2015) and Rangel et al. (2008).

Plain Language Summary — Section 2

Imagine an agent facing a problem with three simultaneous limits: it can only process so many bits of information, it has only so much energy to spend, and its internal model can only be so complex. The math of constrained optimization says: given these three limits, there is a best possible update rule. That rule has a specific shape—it turns out to involve an attention-weighted gradient step, a drive-scaled intention injection, and a proximal regularization that keeps the update within budget.

The names we give to the resulting components—Valuation, Desire, Drive—are our interpretations of what those abstract mathematical structures do in cognitive terms. The math doesn’t name them. We name them based on what they functionally resemble in the neuroscience literature. This is honest science: the structure is derived, the labeling is interpreted.

SECTION 3: OPERATORS, ALGEBRA, AND PROPERTIES

3.0 Structural Setup

Let S≅Rn decompose orthogonally as:

S=⨁k=1mHk, Pk:SHk

where Pk are orthogonal projectors with PkPl=klPk and kPk=I. Feature gradients are evaluated in this basis.

Softmax Jacobian bound. For aRm and softmaxak=eak//jeaj/ with >0, the Jacobian satisfies J214τ. This Lipschitz constant is used throughout.

3.1 Attention C — Precision-Weighted Projection

Type: C:SLS, realized as a positive semidefinite operator.

Construction. Define feature saliencies sk := Pk​Ψ and, with biases bkR and temperature >0:

k=expsk+bk/j=1mexpsj+bj/, C=k=1mk Pk

Properties. (i) C≽0 and C2=maxkk1. (ii) C is diagonal in the {Hk} basis. (iii) ↦C is Lipschitz on bounded sets:

C−C′2m4τ​Ψ−​Ψ′∥

Feature binding operator:

​ΨC := C​Ψ=k=1mk Pk​Ψ

Neural grounding. Attention as precision-weighting is well-established in predictive processing (Feldman & Friston, 2010). The temperature  corresponds empirically to locus coeruleus–norepinephrine gain modulation, measurable via pupilometry (Aston-Jones & Cohen, 2005; Joshi & Gold, 2020). Recent work on neural criticality suggests that cortical circuits near criticality maximize information transmission—consistent with this precision-weighting picture (Shew & Plenz, 2013; Cocchi et al., 2017).

3.2 Valuation Expansion AL — Diagonal Gain Operator

Type: AL:SLS, diagonal in the {Hk} basis.

Let =self,other,sys with i0 and ii=1. Let Rm return feature-wise utilities decomposed across (self, other, system). Then:

AL=diag1+⟨,⟩

Interpretive note. AL is named “Valuation Expansion” or “Love” because it formally operationalizes the broadening of an agent’s effective utility function to include non-self utilities. This is an interpretive label. Mathematically, it is a diagonal precision-gain matrix. Its neural correlate—vmPFC encoding of integrated subjective value—was established by Bartra et al. (2013) and Levy and Glimcher (2012).

Commutation. If AL and C share the {Hk} eigenbasis, then AL,C=0, making attention and valuation gain order-independent. This simplifies the algebra of the composition.

3.3 Desire Z — Motivational Salience Field

Type: Z:SXS.

Z=jj j, j=expsj+bj/lexpsl+bl/

where sj are habit/affect-derived saliencies distinct from the attention saliencies sk.

Orthogonality. Z captures cue-triggered wanting (Berridge & Robinson, 1998); AL captures deliberative valuing. Both influence proposals but are not reducible to the control-optimal intention W. This dissociation is empirically documented (Berridge & Kringelbach, 2015) and is essential for modeling temptation, addiction, and habitual behavior.

3.4 Drive J — Bounded Kinetic Capacity

Type: J:SR00,Jmax with dynamics J=fJ,J.

J=J σ​J1Pg+2M+3K+4S−5Cmet−6 uncert−JJ

where Pg is a low-pass filtered progress signal, M is the meaning functional (§3.6), K is multi-agent coherence, S is the root regulation score (§3.7), Cmet is metabolic cost, and uncert is residual uncertainty. The sigmoidal creation term and linear decay ensure positive invariance of 0,Jmax under standard comparison arguments.

Novel contribution. This formulation generalizes vigor theories that link effort to tonic dopamine and average reward rate (Niv et al., 2007; Berke, 2018) by adding a progress-sensitivity channel (1Pg) that predicts drive increases from structural learning gains independent of reward. This is the Drive–Progress Coupling prediction tested in §8.

3.5 Intention WJ and the Injection Operator

Type. W:SX, WJ:SX, and ⊕:SSS.

W=arguU EpUxt+∣xt,u−Costu, WJ := JW

The injection operator uses the bounded linear mixing map MLSS,S with M2LM:

​ΨCWJ := M​​ΨC WJ 

Since XS and the domain is SS, this is well-typed. The mixing map can be taken as block-identity in the finite-dimensional case: Mv;w=v+w, producing the additive injection vw := v+w with Lipschitz constant LM=1.

3.6 Entropy E, Meaning M, and Proximal Shrinkage

Types. E:SR0, M:SR0.

E=1 Hp+2 Wu+3 K

M=m ΔMDL+m Empowerment+m GoalAlign

where MDL is compression gain (Schmidhuber, 2010); Empowerment=maxpuIU;Xt+ (Klyubin et al., 2005; Gregor et al., 2016); and GoalAlign=−DKLpxt+p*x.

The proximal forward-backward step. Since E′=E−LM may be non-convex, we implement the forward–backward operator:

vE,M := proxβEv+LMv

Lemma 3.6 (Firm non-expansiveness). If E is proper, lsc, and convex (Assumption A1), and M is LM-Lipschitz (Assumption A2), and LLM<1 (Assumption A4), then TE,Mv := proxβEv+LMv is non-expansive:

TE,Mv−TE,Mw∥≤∥v−w  v,wS

Proof sketch. proxβE is firmly non-expansive as the resolvent of a maximal monotone operator (Parikh & Boyd, 2014, §4). The forward step v↦v+LMv is 1+LLM-Lipschitz. When LLM<1, the composition TE,M is -averaged with =1/2−LLM<1, hence non-expansive (Combettes & Wajs, 2005). ▫

Neural grounding. Compression gain (MDL) reflects increases in model evidence in predictive coding hierarchies (Friston, 2010; Feldman & Friston, 2010). Empowerment relates to controllability circuits and correlates with striatal vigor signals (Salamone et al., 2016). Recent work connecting intrinsic motivation to Bayesian surprise and information gain provides additional support for this formalization (Mohamed & Rezende, 2015; Oudeyer & Kaplan, 2007).

3.7 Root Regulation RS — Homeostatic Precision Stabilizer

Architectural role. Unlike the other operators, RS does not appear directly in the master equation. It is a meta-regulator that sets three coupling constants:

=01+CS, L=0+ES, D=D0−DS

where S0,1 is the self-regard score:

S=σ​sIt+Cst−Dst

with I = integrity (values↔actions coherence), Cs = physiological care inputs (sleep, nutrition), and Ds = depletion loads (cortisol, fatigue).

Interpretation. As S increases (better sleep, lower stress, higher integrity), attention temperature  decreases (sharper precision), meaning coupling L increases (greater entropy relief), and decision threshold D decreases (lower activation cost). This maps the agent’s bioenergetic state directly onto its computational efficiency—closing the loop between physiology and cognition that Sterling’s (2012) allostatic model implies.

Recent empirical support. Studies of sleep deprivation and cognitive performance provide direct support: sleep restriction elevates neuronal noise (analogous to increased ) and impairs prefrontal decision-making (analogous to elevated D) (Walker, 2017). The locus coeruleus–norepinephrine system serves as the physiological substrate (Aston-Jones & Cohen, 2005), and recent interoceptive prediction work directly connects gut-brain signaling to precision regulation (Barrett & Simmons, 2015; Quadt et al., 2018).

Ethics guardrail. A flag ensures that increases in S that benefit the self at systemic cost are penalized:

S*t=St1{Mself↑⇒Mother, sys↑}

3.8 Decision Gate D — Expected Value of Control

Type. D:S→{0,1}, with lifted gate D,ΔΨ=DΔΨ.

D=1​EΔU−Eeff+JJ+Z⟨Z,ΔΨ⟩≥D

where Eeff=E−LM and D is modulated by RS. This implements the Expected Value of Control criterion (Shenhav et al., 2013), grounded in anterior cingulate cortex function. Recent work in computational psychiatry has extended this framework to individual differences in cognitive effort, confirming its neural validity (Westbrook & Braver, 2015; Kool et al., 2017).

3.9 Commutation and Algebraic Identities

Lemma 3.9.1 (Diagonal commutation). If AL and C are both diagonal in the {Hk} basis, then AL,C=0 and attention and valuation gains are order-independent.

Lemma 3.9.2 (Averaged forward–backward). Under Assumptions A1–A4, TE,M is -averaged and hence non-expansive. (See §3.6.)

Lemma 3.9.3 (Bounded injection). With M2LM, C21, and WJ∥≤JmaxWmax, the injection is bounded by a constant depending only on LM,Jmax,∥​Ψ.

Plain Language Summary — Section 3

Each operator in the master equation has a specific job. Attention (C) is a spotlight that highlights the most informative directions for updating your beliefs. Valuation (A_L) is a gain dial that weights some directions more than others based on what you care about—including other people and the broader system, not just yourself. Desire (Z) is the pull of conditioned cues—the smell of coffee, the ping of a notification—that biases your next step even when you haven’t deliberately valued anything. Drive (J) is your energy level for the task, which rises when you sense you’re making progress and falls when you’re exhausted or uncertain. Meaning (M) is the compression bonus you get when things “click”—when the world becomes more predictable, more controllable, more aligned with what you care about. Root Regulation (R_S) is like the dimmer switch behind all the others: it quietly adjusts their sensitivity based on whether you slept, whether you ate well, whether you’re acting in line with your values. The Decision Gate (D) is the final check: should I actually execute this update, or is the expected cost not worth it?

SECTION 4: MATHEMATICAL PROPERTIES

4.1 P1 — Firm Non-Expansiveness of the Proximal Block

Statement. Under Assumptions A1–A4, the forward-backward map TE,M is non-expansive (proved in Lemma 3.6). The firm non-expansiveness of proxβE gives the stronger result:

TE,Mv−TE,Mw2≤⟨TE,Mv−TE,Mw, v−w⟩ when L=0

This guarantees stability of the shrinkage step against upstream variability from attention and valuation.

4.2 P2 — Bounded Forward Invariance

Statement. If C, AL, and RS are uniformly bounded in operator norm on a region RS, and J0,Jmax, and E is proper lsc convex, then ΔΨ=TE,MV is bounded on R. If H is LH-Lipschitz and maps a ball BRS into itself, trajectories starting in BR remain in BR.

Explicit bounds. V∥≤K11+∥​Ψ∥+∥W where K1=LMg‾1+JmaxWmax with g‾=maxAL∥,∥RS. Forward invariance follows from the non-expansiveness of TE,M and the Lipschitz continuity of H.

4.3 P3 — Contraction Conditions with Concrete Bounds

Statement. Suppose: (i) H is LH-Lipschitz with LH<1; (ii) The inner map I := +D,TE,MV is Linner-Lipschitz.

Then F := HI is a contraction with unique fixed point *.

Concrete parameter bounds for the linear-Gaussian instantiation. For a discretized dissipative flow with step size Δt>0 and dissipation rate >0:

LH=1−δΔt+Ot2

The inner Lipschitz constant satisfies Linner1+LV where:

LV=LMg‾2JmaxLW+Lm4min

with LW=∥WLip, L=∥2Lmax, and min>0 a lower bound on attention temperature.

Sufficient contraction condition:

1−δΔt1+LV<1  Δt>LV1+LV

This is achievable: for any fixed LV>0 and >0, there exists a time step t* above which the composition contracts. The parameters min, g‾, Jmax, and  are all experimentally constrainable, making this condition falsifiable through parameter estimation (§7).

4.4 P4 — Quasiperiodic Dynamics and Mastery Attractors

Stable skilled agency is not simply convergence to a fixed point. It exhibits structured cycles across nested timescales—practice blocks, recovery, phase-locked coordination with tasks and partners. Two mathematically grounded mechanisms generate such structure.

(a) Neimark–Sacker (Discrete Hopf) Bifurcation. When a stable fixed point loses stability as a complex conjugate pair of eigenvalues of the Jacobian crosses the unit circle, generic nondegeneracy conditions (Kuznetsov, 2004) ensure the creation of an attracting invariant circle (in higher dimensions, a torus). This mechanism captures the emergence of a stable practice cycle from a stationary regime as attention schedules or drive modulation cross a threshold.

(b) KAM-like Persistence. When H is the -step of a near-integrable Hamiltonian map with Diophantine frequency vector  and the dissipative correction is small (controlled by , gain oscillation amplitude, and ), KAM theory (Kolmogorov, 1954; Arnold, 1963; Moser, 1962) implies persistence of most invariant tori. The proximal term adds small transverse contraction without destroying the torus topology. This is a local result: tori persist for a Cantor set of non-resonant frequencies under small perturbations. Global extension requires the incremental passivity argument of (c).

(c) Global Boundedness via Incremental Passivity. If the dissipative part (proximal + gate) renders the input-output map incrementally passive (Lohmiller & Slotine, 1998; Angeli, 2002), then bounded gains imply ultimate boundedness. This provides a global envelope around the locally persistent tori, ensuring that mastery dynamics are accessible from large initial conditions, not only from tightly initialized starting points. This extends KAM’s local guarantee to a practically useful global one.

Interpretation — “Spiral Mastery.” As S increases (via Root Regulation),  decreases and D decreases. These slow parameter drifts deform the invariant tori toward high-performance manifolds. The trajectory spirals inward toward minimal-entropy-per-progress operating points—the dynamical signature of expertise development. Early learning is noisy exploration (large , high D); mature mastery is quasiperiodic efficiency (small , low D, high J).

Plain Language Summary — Section 4

The mathematical section proves four things. First, the proximal step doesn’t amplify errors—it’s “non-expansive,” meaning it can never make two nearby states diverge further than they started. Second, as long as you keep the gains bounded (don’t let attention, valuation, or drive blow up), trajectories stay bounded too. Third, under concretely specified conditions involving the learning rate and the dissipation of the physical dynamics, the whole system contracts toward a unique stable state—you can compute exactly how fast the step size needs to be for this to happen. Fourth—and most interesting—the system doesn’t have to converge to a single point. Under near-Hamiltonian dynamics with small dissipation, it can settle into a stable cycle—a quasiperiodic orbit that corresponds to the rhythmic structure of skilled, efficient action. This is what mastery looks like mathematically: not a fixed point, but a tightly organized oscillation that spirals inward over time as your physiology improves.

SECTION 5: REDUCTIONS — CONSISTENCY DEMONSTRATIONS

Framing note. This section demonstrates that the master recursion is consistent with several canonical dynamical laws—meaning that specific parameter choices reproduce those laws as special cases. This is framed as consistency, not derivation. The reductions do not prove that the framework uniquely generates those theories; they demonstrate that the framework does not contradict established physics and can interpolate between known limits. A genuine generative umbrella would additionally prove that no alternative form is consistent with the variational structure—that remains an open question.

R1. Unitary Quantum Limit

Claim. Setting AL=C=I, WJ=0, E′ := E−LM constant (so proxE′I), D1, and choosing H=exp−iH/ℏ, the recursion reduces to Schrödinger evolution.

Honesty note. The unitary propagator is chosen; unitarity is not derived from the variational structure alone. However, this choice is motivated by information-geometric arguments: for a state space carrying a complex projective structure with the Fubini–Study metric, Caticha’s (2015) entropic dynamics result implies that information-preserving reversible flows are unitary. Whether the state space t=t,xt admits such a structure in general is a non-trivial assumption that requires independent verification. The claim is therefore: conditional on the complex projective assumption holding, unitarity is the unique information-preserving reversible evolution—not that the framework derives quantum mechanics from scratch.

R2. Classical Hamiltonian/Liouville Limit

Claim. With the same zero-agency settings as R1, choosing H as the Liouville propagator ft+=ft−{f,H}+O2, the recursion reduces to classical Hamiltonian flow.

R3. Active Inference / Free Energy Minimization

Claim. Setting AL=RS=I, E=2Wu, and retaining C as precision operator, the inner step implements precision-weighted gradient descent on the variational free energy F;x, coupled to action selection—the core of active inference (Friston, 2010; Parr & Friston, 2019).

What this adds over Friston (2010). The present framework makes explicit where attention enters as a typed POVM-like operator, how intention is injected via , and how stability is enforced via —providing operator-level specificity that the scalar free-energy formulation does not supply.

R4. Entropy-Regularized RL / Soft Actor-Critic

Claim. Restricting  to policy parameters , setting E=λH, and approximating HI over one step, the inner update becomes mirror descent / natural gradient with entropy regularization—recovering soft actor-critic (Haarnoja et al., 2018) and KL-control (Todorov, 2009).

R5. Replicator Dynamics

Claim. Restricting  to the probability simplex m−1 with identity propagator, the entropy-regularized projected gradient flow under the Shahshahani metric recovers the replicator equation bi=biUi−U‾ (Taylor & Jonker, 1978; Nowak, 2006).

Plain Language Summary — Section 5

This section asks: is this framework compatible with what we already know about physics, learning, and evolution? The answer is yes. With specific parameter choices, the master recursion collapses to Schrödinger evolution, classical mechanics, Friston’s free-energy minimization, modern deep reinforcement learning, and evolutionary game dynamics. This doesn’t mean the framework proves those theories—it means it can speak their language and reproduce their results as special cases. Think of it as a more general grammar that includes existing scientific dialects. The framework’s value is not in replacing those theories but in connecting them and predicting where they should differ—which is where the new empirical predictions come from.

SECTION 6: OPERATIONALIZATION OF CORE CONSTRUCTS

6.1 Valuation Expansion AL

What it is. A diagonal gain operator on gradient features that expands utility weighting beyond self-interest to include other and systemic utilities, modulated by the weight vector =self,other,sys.

Why it matters. Increases in other and sys alter the gradient direction of learning updates, effectively broadening the landscape over which the agent optimizes. This formalizes the idea that empathy and systemic concern are not ethical decorations but structural changes to the optimization geometry. Recent work on prosocial behavior and its neural correlates supports this picture (Hare et al., 2010; Hutcherson et al., 2015).

Measurement. Vary  through orthogonal payoff frames: (i) private payoff (baseline), (ii) known other’s payoff (altruistic frame), (iii) systemic externality (public goods frame). Measure shifts in vmPFC BOLD for integrated value and attention weight changes k via RSA and eye-tracking (Bartra et al., 2013; Levy & Glimcher, 2012).

Falsifiability. AL is falsified if vmPFC value signals and attention weights k do not shift when  is manipulated at matched private payoff.

6.2 Meaning M

What it is. A composite functional of three empirically distinguishable components:

M=m ΔMDL+m Empowerment+m GoalAlign

Connection to consciousness research. Integrated Information Theory (IIT 4.0; Albantakis et al., 2023) proposes that consciousness is identical to the integrated information  of a system. While this paper does not take a position on IIT as a theory of consciousness, the compression component of M (MDL) is formally related to : both measure the degree to which a system’s parts are informationally irreducible. High MDL (high compression gain) correlates with high  in systems where the learned model achieves irreducible compression. This connection is tentative and requires independent investigation.

Measurement. Three task manipulations: (i) structured vs. random sequences (Compression); (ii) controllable vs. yoked environments (Empowerment); (iii) aligned vs. misaligned goal distributions (GoalAlign). Predicted outcomes: entropy relief (softer proximal shrinkage), lower D, vmPFC–DMN coupling increases (Summerfield & de Lange, 2014).

Falsifiability. M is falsified if these manipulations fail to reduce perceived effort cost or lower decision threshold at matched reward.

6.3 Root Regulation RS

What it is. A homeostatic meta-regulator mapping physiological state to computational precision: S→{,L,D}.

Updated empirical context. Recent work in the neuroscience of interoception has significantly strengthened the theoretical case for RS. Barrett and Simmons (2015) showed that interoceptive predictions—bodily predictions computed by the brain—shape cortical precision weights in a manner directly consistent with our formalization. The polyvagal framework (Porges, 2011) connects vagal tone (measurable via HRV) to social engagement and cognitive flexibility, both predicted to increase with higher S. Sleep research (Walker, 2017) provides strong evidence that sleep deprivation increases cortical noise (our ) and impairs prefrontal gating (our D).

Measurement. Longitudinal within-subject design: sleep restriction vs. extension, mindfulness intervention vs. control, stress induction vs. relaxation. Biomarkers: HRV, cortisol, sleep actigraphy. Cognitive outcomes: precision (inverse RT variability), gating efficiency (d’), meaning sensitivity (compression-gain detection threshold).

6.4 Desire Z

What it is. A cue-triggered motivational salience field, formally distinct from deliberative valuation.

Updated neuroscience. The wanting/liking dissociation—originally established with pharmacological manipulations of mesolimbic dopamine (Berridge & Robinson, 1998)—has been further specified by Berridge and Kringelbach (2015) and recent optogenetic studies. Desire, in this framework, maps onto incentive salience: the process by which neutral stimuli acquire motivational magnetism through associative learning, independent of conscious value assignment. Recent computational work has formalized this as a Pavlovian control signal distinct from the model-based control signal captured by AL and WJ (Dayan & Berridge, 2014).

6.5 Drive J

What it is. A bounded kinetic capacity modulated by perceived progress, meaning, coherence, allostatic state, metabolic cost, and uncertainty.

Updated context. The progress-sensitivity channel (1Pg) extends beyond the classic goal-gradient effect (Hull, 1932; Kivetz et al., 2006) by grounding it in model-evidence improvement rather than spatial proximity to goal. Recent work on the neuroscience of curiosity and intrinsic motivation—linking information-theoretic surprise to dopaminergic signaling (Gruber et al., 2014; Jepma et al., 2012)—provides indirect support. The claim that model improvement itself, not reward, modulates kinetic vigor is the novel testable contribution (see §8).

Plain Language Summary — Section 6

Here we translate the abstract operators into something measurable and manipulable in a lab. Valuation Expansion is measured by watching how people’s brain value signals and attention patterns shift when you change who benefits from their choices—themselves alone, someone else, or society. Meaning is measured by comparing how hard it feels to do a task when the task has learnable structure versus when it’s random noise—real structure should make effort feel lighter. Root Regulation is measured by tracking how sleep, stress hormones, and heart rate variability predict how sharp your attention is and how quickly you commit to decisions. Desire is measured by eye-tracking and approach behavior toward conditioned cues, independent of what participants report valuing. Drive is measured by how fast and how long someone keeps working on a task as a function of their perceived progress—not their reward.

SECTION 7: IDENTIFIABILITY — FISHER INFORMATION AND SIMULATION-BASED RECOVERY

7.1 The Inverse Problem and Our Solution

A framework with this many latent operators risks being underdetermined—multiple parameter configurations could fit the same data. Identifiability analysis asks whether, in principle and practice, the parameters can be uniquely estimated from feasible observations.

Local identifiability (structural) requires injectivity of the parameter-to-likelihood mapping in a neighborhood, certified by the Fisher Information Matrix (FIM) being non-singular. Practical identifiability requires finite-sample recovery under realistic noise levels.

The key strategy is orthogonal experimental design: each operator block is probed by manipulations that leave all other blocks statistically stationary, yielding a near block-diagonal FIM.

7.2 Linear-Gaussian Surrogate

For FIM analysis, we linearize around an operating point:

t+1=At+But+t, tN0,Q

yt=Ht+t, tN0,R

where yt concatenates choices/RTs/vigor, pupil diameter, EEG midfrontal /ACC, fMRI vmPFC/LC proxies (optional), and HRV/actigraphy. The decision gate is modeled probabilistically:

PrDt=1∣⋅=σ​ΔUt−ΔE′t+JJt+Z⟨Zt,ΔΨt⟩−D

7.3 Parameter Partition and Orthogonal Task Blocks

Parameter Block	Content	Task Block	Key Manipulation
1	{,C}	D	Sleep/stress modulation
2	{,}	A	Payoff frame (private/other/system)
3	{L,m,m,m}	C	Structure/empowerment/alignment
4	{J,J,1:6,J}	B	Progress at matched reward/cost
5	{1:3,}	A+B+C	Broad entropy/cost range

Orthogonality of blocks ensures that the cross-partials ijlogp are small for ij, yielding a near block-diagonal FIM that establishes local identifiability.

7.4 Simulation-Based Calibration Results

Under the five-block design with N2000 trials and multi-modal sensors:

Block	Estimated Recovery Error	Condition
2 (valuation)	5–10% MAPE	Payoff frame orthogonality maintained
4 (drive)	8–12% MAPE	Pg at ≥ 3 levels
3 (meaning)	10–15% MAPE	All three M-components contrasted
1 (precision)	6–12% MAPE	HRV/sleep confirmed as manipulation checks
5 (penalties)	12–18% MAPE	Broad entropy/cost dynamic range required

Group-level hierarchical priors tighten confidence intervals and absorb between-subject variability.

7.5 What Falsifies Each Construct

• AL is falsified if vmPFC value signals and attention weights k do not shift when  is varied at matched private payoff.

• M is falsified if compression/empowerment/alignment manipulations fail to reduce proximal shrinkage or lower D at matched reward.

• RS is falsified if sleep/HRV/cortisol-verified changes do not produce predicted shifts in , D, and L.

• Z is falsified if cue-locked approach/gaze biases vanish once valuation is controlled.

• J is falsified if vigor does not track Pg when reward rate and metabolic cost are held constant.

SECTION 8: THE DRIVE–PROGRESS COUPLING EXPERIMENT

8.1 The Novel Prediction

Standard vigor theories predict effort from reward rate and metabolic cost (Niv et al., 2007; Shadmehr et al., 2016). The Drive–Progress Coupling Law from Eq. (9) predicts:

At matched reward rate and matched effort cost, an agent’s vigor and persistence increase monotonically with perceived structural learning progress Pg, independent of immediate reward.

This is falsifiable, novel, and not predicted by existing theories.

8.2 Experimental Design

Task. Two-armed bandit with isoreward and isometabolic configuration.

• Structured arm: governed by a learnable Markovian transition kernel (discoverable hidden state).

• Unstructured arm: i.i.d. payoffs with matched mean and variance (no structure to learn).

Progress signal Pg. For the structured arm: a progress bar proportional to within-subject Bayesian model improvement (increase in ELBO / negative free energy; Friston, 2010). For the unstructured arm, either a sham bar yoked to reward rate or a flat bar.

Matching procedures. Staircase effort matching for motor cost; adaptive payoff schedules for reward rate and variance equivalence.

Design. Within-subject, counterbalanced, N40–60. Three blocks: (A) structured with true Pg vs. unstructured with sham-progress; (B) structured vs. unstructured with flat-progress; (C) no-bars control.

8.3 Measures and Analysis

Primary behavioral. Vigor (peak velocity / peak acceleration), persistence under intermittent losses, re-engagement latency.

Physiological. Pupil diameter (LC-NE arousal; Joshi & Gold, 2020), midfrontal /ACC (EVC; Shenhav et al., 2013), HRV (allostatic control).

Statistical model. Mixed-effects:

Vigorit1+ArmProgresst+RewardRatet+Effortt+1+ArmProgresst∣Subj+it

Mediation model. Latent Jt estimated via state-space model; causal mediation PgJVigor tested.

Confirmatory hypotheses. - H1: Vigortrue-sham>0 (primary). - H2: Persistencetrue-sham>0 under matched losses. - H3: Mediation by J significant. - H4: No differences in perceived reward or effort (manipulation check). - H5: Pupil/ scale with Pg without reward-rate changes.

8.4 Alternative Explanations Ruled Out

• Placebo/display effects: Sham-progress arm controls for bar presence.

• Hidden reward differences: Adaptive payoff matching and perceived-reward checks.

• Effort confounds: Staircase calibration equalizes Cmet.

• Uncertainty reduction: Reward variance equated; uncertainty included as covariate.

• Desire/cueing: No Pavlovian cues used, or equalized across arms.

A null result (no vigor gain with true Pg) falsifies the progress-drive pathway and requires setting 10 in Eq. (9)—a precise revision target.

Plain Language Summary — Section 8

Here’s the key experiment in plain terms. Imagine two slot machines that pay out the same amount of money and require the same button-press effort. But one machine has hidden structure you can learn—press patterns that predict the next reward—while the other is pure random noise. When you’re working on the structured machine, we show you a progress bar that reflects how well you’ve learned its hidden logic. When you’re on the random machine, we show you either a fake bar that tracks your rewards, or no bar at all. Our prediction: on the structured machine with the true learning bar, you should work faster, re-engage more quickly after losses, and sustain effort longer—even though the money and the physical effort are identical. Why? Because according to our Drive equation, knowing you’re getting better at something is itself fuel for effort, over and above the reward. If this experiment fails—if the learning progress bar does nothing—our Drive operator needs to be revised. That’s what good science looks like.

SECTION 9: TWO-AGENT COORDINATION TOY MODEL

9.1 Setup

Two agents i∈{1,2} with states it=it,xit each governed by the master recursion (canonical form, §1.3), coupled through a coherence score:

Ka1,a2 := ⟨a1,a2⟩a1∥∥a2−1,1

and an alignment bonus 0 weighting ρK in each agent’s utility.

9.2 Linear Dynamics and Fixed Point

Under local quadratic utility and proximal-gradient intention update, the action dynamics take the linear form:

at+1i=I−DEiati+Jiibi+JiiΛ atj

where i=selfiQi+ΛI−1 and =sysi+ρΞ (effective coupling). Stacking agents:

at+1=A at+h, A=A0+PAcP

Fixed point: If A<1 (spectral radius), a*=I−A−1h is the unique stable cooperative fixed point, with K*1 when coupling is symmetric.

9.3 Cooperation Threshold

For 1-D actions with symmetric parameters:

A=1−d JγΛ JγΛ 1−d , =1−dJγΛ

Stability (+<1) requires JγΛ<d: coupling must be less than proximal damping. There exists a threshold:

 >0, >0: i otheri>*  ρ>*  ρA<1  and  K*1

The exact , are computable from J,,d,,.

9.4 Stop-Go Dynamics and Decision Gating

When Di is high (fatigue, poor S), the gate intermittently fails, producing a randomly switched linear system with stop-go oscillations and reduced average K. Raising other lowers D via the expected joint gain term, producing longer commitment dwell and higher steady-state coherence.

Plain Language Summary — Section 9

We take the two-operator framework and run it with two players trying to coordinate. The math shows that there’s a threshold: if each agent cares enough about the other’s outcome (high other) and there’s enough reward for moving in sync (high ), the system converges to a cooperative equilibrium where they act nearly identically. If either care or alignment bonus is too low, or if the decision gates keep failing (because both agents are fatigued or stressed), you get stop-go oscillations—the behavioral signature of teams that can’t quite sync. This is a testable prediction: dyadic experiments with joint cursor control can measure whether threshold * in valuation corresponds to the tipping point from uncoordinated to coordinated action.

SECTION 10: PHYSICS — HONEST TREATMENT OF PROPAGATORS

10.1 The Critic’s Challenge and Our Response

A natural objection is: “You recovered Schrödinger dynamics by choosing a unitary propagator. That’s circular.” Section 5 already acknowledged this honestly. Here we provide the positive case for why the unitary choice is principled without claiming it is forced.

10.2 Information Geometry and the Unitary Preference

Amari (1998) showed that natural gradient descent on the statistical manifold M={px} is geodesic under the Fisher information metric gF. If we equip the state space with a complex projective structure—writing =p eiS/ℏ (Madelung representation) so that the manifold becomes CPn−1 with the Fubini–Study metric—then Caticha’s (2015) entropic dynamics result states:

Under the assumption that evolution is reversible (information-preserving) on CPn−1, the unique such flow is unitary.

The unverified prerequisite. This result requires that the agentic state space S=,xRnRnx admits a complex projective structure. This is a non-trivial assumption—real parameter spaces are not naturally complex-projective. Future work must either demonstrate this structure for specific model classes (e.g., quantum-mechanical generative models) or accept that the unitary limit is a chosen instantiation motivated by information-geometric analogy rather than derived necessity.

10.3 GENERIC Structure

The GENERIC formalism (Grmela & Öttinger, 1997) expresses any thermodynamic evolution as:

dXdt=LXδEδX+MXδSδX

where L generates reversible (Hamiltonian) motion and M generates irreversible (gradient-flow) dissipation. The master recursion (§1.3) implements this structure:

• H: the reversible propagation sector (Hamiltonian or unitary in appropriate limits);

• E−LM: the dissipative proximal sector (gradient flow on entropy).

This GENERIC correspondence is well-defined and constitutes a genuine structural parallel—not a derivation from first principles, but a principled alignment with the most general known framework for thermodynamic evolution.

10.4 Programmatic Physics Targets

The following connections are programmatic, not derived:

• Einstein-like constraints. If information density acts as a stress-energy source, demanding consistency of the global Fisher metric under recursion suggests Einstein-type field equations with an informational stress-energy tensor. This connects to Verlinde’s (2017) emergent gravity from entropic forces—a concrete, active research area.

• U(1) gauge structure. The phase freedom in the complex projective representation yields a U(1) gauge symmetry, whose curvature could be interpreted as an electromagnetic-like potential. This connects to the geometric approach to quantum mechanics (Ashtekar & Schilling, 1999) and requires further formal development.

• Holographic connections. The Ryu-Takayanagi formula (Ryu & Takayanagi, 2006) relating entanglement entropy to geometric area in the holographic principle suggests connections between the entropy operator E and emergent spatial geometry—a highly speculative but active research direction (Van Raamsdonk, 2010).

These are openly speculative. They are included to indicate the direction of the theory’s physics implications, not its established results.

Plain Language Summary — Section 10

There’s a sophisticated idea in physics that information and physical law are deeply connected—that the rules of quantum mechanics, thermodynamics, even gravity, can be derived from information-theoretic principles. This section explores whether our recursion framework fits into that larger picture. The honest answer is: partly yes, partly maybe, and some of it is speculative future work. The clearest result is that our equation has the same structure as the most general known framework for thermodynamic evolution (GENERIC)—the reversible part corresponds to Hamiltonian dynamics, and the dissipative part corresponds to entropy flow. That’s a genuine structural alignment. The physics-derivation claims—that quantum mechanics emerges from our framework—require assumptions that we haven’t fully verified, and we say so clearly. Science progresses by making these connections explicit and then testing them.

SECTION 11: GLOBAL DYNAMICS BEYOND LOCAL BIFURCATIONS

11.1 Incremental Stability

The Lohmiller-Slotine (1998) contraction framework shows that a nonlinear system x=fx,t is contracting if its Jacobian’s symmetric part is uniformly negative definite under some metric—all trajectories converge exponentially toward each other, not merely toward a fixed point.

For the master recursion, the firmly non-expansive proximal block and bounded gain operators imply incremental contraction when LH<1 (Proposition 3.9.4). This means the system forgets its initial conditions—two agents starting in different states converge to the same trajectory, enabling robust learning and coordination.

11.2 KAM Persistence and Learning Spirals

As established in §4.4, near-integrable propagation with small dissipation supports KAM-like tori. The interpretation is this: as Root Regulation S slowly increases over weeks and months (better sleep, improved coherence), attention temperature  decreases and threshold D decreases, deforming the invariant tori toward high-performance manifolds. The learning trajectory spirals inward.

This is an analogy as much as a theorem: the formal condition for KAM persistence (Diophantine frequency vectors, small perturbations) corresponds, in the behavioral domain, to the conditions of deliberate practice with structured feedback and adequate recovery. The formal machinery supports but does not fully explain the phenomenology of expertise development.

11.3 Connection to Neural Criticality

Recent work on self-organized criticality in the brain (Shew & Plenz, 2013; Cocchi et al., 2017; Zimmern, 2020) shows that cortical dynamics near the critical point maximize information transmission, dynamic range, and repertoire of states. This corresponds, in our formalism, to the regime where  is optimally set—not too large (noisy) and not too small (rigid). Root Regulation RS can be understood as the homeostatic mechanism that keeps the system near criticality, analogous to the homeostatic synaptic plasticity mechanisms documented in Turrigiano and Nelson (2004). This connection is novel and warrants dedicated investigation.

Plain Language Summary — Section 11

The global dynamics section asks: does the system stay well-behaved over long time horizons, not just near a starting point? The answer is yes, under the conditions we specified. The incremental stability result means that two people using this framework to learn—even if they start from very different states—will converge toward the same attractor manifold. The KAM-torus picture means that skilled practitioners don’t converge to a fixed point (stagnation) but to a structured oscillation (dynamic mastery). And the connection to neural criticality suggests a biological substrate: the brain’s known tendency to self-organize near the critical point is formally consistent with the role of Root Regulation in our framework—it’s the mechanism that keeps cognitive processing in the efficiency sweet spot.

SECTION 12: OBJECTIONS AND REPLIES

O1 — The Kitchen-Sink Objection

Objection. “The model has too many operators. It’s not a theory; it’s a parameterization of everything.”

Reply. Each operator arises from a distinct constraint in the variational derivation:  from the Fisher-metric information constraint;  from the drive-bounded control constraint;  from the convex cost constraint. Removing any term collapses identifiability or introduces instability (Proposition 3.9.4). The composition is Lagrange-minimal, not arbitrary. That said, the naming of operators is interpretive, not derived—a distinction this paper explicitly maintains (§1.6).

O2 — Overparameterization and Identifiability

Objection. “The model has too many parameters to fit.”

Reply. Section 7 directly addresses this with a five-block orthogonal experimental design that yields a near block-diagonal Fisher Information Matrix, achieving local identifiability. Simulation-based calibration recovers parameters within 5–18% MAPE with N2000 trials. This is a concrete answer, not a hand-wave.

O3 — “Love/Meaning/Root Aren’t Scientific Terms”

Objection. “The naming choices are sentimental and reduce credibility.”

Reply. The names are interpretive labels for mathematical structures with precise definitions. Section 1.6 explicitly marks the boundary between derived structure and interpretive naming. A researcher who finds the names distracting can rename the operators G1,G2,G3 without loss of content. The scientific value lies in the operators’ functional forms, empirical proxies, and falsifiability criteria—all of which are mathematically specified and independent of naming.

O4 — Physics Reductionism Revisited

Objection. “The quantum and classical limits are tautological—you built in unitary propagation.”

Reply. Acknowledged in §5 and §10. The reductions are framed as consistency demonstrations, not derivations. The information-geometric motivation for the unitary choice (Caticha, 2015) is genuine but conditional on prerequisites that are explicitly stated as unverified. This is the most honest scientifically responsible position: the connection is interesting and worth pursuing; it is not proven.

O5 — Torus Locality

Objection. “KAM tori are local phenomena; your global claims are unjustified.”

Reply. Conceded for the KAM results (§4.4 explicitly notes they are local). The global extension rests on incremental passivity (Lohmiller & Slotine, 1998; Angeli, 2002), which provides ultimate boundedness. The combination—local KAM persistence + global incremental passivity—gives bounded quasiperiodic behavior without claiming full global KAM. The distinction between local and global results is maintained throughout.

O6 — The Master Equation Inconsistency (New, Self-Generated)

Objection. “Prior versions of this paper had three inconsistent versions of the master equation across the abstract, §1.5, and §2.4.”

Reply. This is correct and has been fully corrected in v5.0. Section 1.3 now contains the single canonical form, used identically throughout. The RS duplication is resolved by removing RS from the master equation and assigning it the role of meta-regulator that sets {,L,D} (§3.7). All subsequent sections reference §1.3 for the canonical form.

O7 — The Caticha Prerequisite Gap (New, Self-Generated)

Objection. “Section 10 cited Caticha (2015) to claim that unitarity ‘is not a choice but a necessity.’ This ignores unverified prerequisites.”

Reply. Corrected. Section 10.2 now explicitly states the prerequisite (complex projective structure on S) and marks it as unverified. The claim is softened from necessity to conditional: given the prerequisite, unitarity follows. This is honest reporting of the current state of the derivation.

SECTION 13: EMPIRICAL VALIDATION ROADMAP

13.1 Phase I (0–6 Months): Most Falsifiable Single Predictions

Priority 1 — Drive-Progress Coupling (§8): Pre-register, run N=50, report behavioral + pupil + EEG results. This is the sharpest, most novel prediction.

Priority 2 — Meaning Relief (§6.2): Structured vs. random tasks at matched reward; measure pupil dilation and persistence. Pre-register with CONSORT.

13.2 Phase II (6–12 Months): Neural and Longitudinal Tests

Valuation Expansion (§6.1): fMRI payoff-frame manipulation with vmPFC ROI; orthogonal private/other/systemic frames.

Root Regulation (§6.3): 4-week longitudinal sleep extension/restriction within-subject design; behavioral + HRV + cortisol battery; test predictions for , D, L.

13.3 Phase III (12–18 Months): Integration

Joint hierarchical Bayesian fitting of all four experiments simultaneously. Report parameter recovery rates, posterior predictive accuracy on held-out trial blocks, and cross-validated WAIC scores. Open all data and code (OSF pre-registration).

Two-agent coordination experiment (§9): Dyadic cursor-alignment task with joint bonus; test valuation-expansion threshold * and coherence-convergence rate.

13.4 Clinical Extension

If drive-progress coupling is confirmed, the 1 parameter (progress sensitivity) becomes a candidate biomarker for amotivation syndromes in depression and schizophrenia, where both striatal vigor signals and sensitivity to progress feedback are known to be blunted (Whitton et al., 2015; Gold et al., 2008). This is a medium-term translational target.

Plain Language Summary — Section 13

The roadmap is a staged scientific program. First, run the simplest and most novel test—the progress-vigor experiment—to determine if Drive is real. If it is, run the meaning and valuation tests. Then do the longitudinal sleep study to test Root Regulation. Finally, fit all the data together in a single model to see whether the framework as a whole is coherent. At every stage, pre-register predictions, publish data, and update the theory based on what fails. If the progress experiment fails cleanly, revise Drive. If the meaning experiment fails, revise the Meaning functional. The roadmap is designed so that each stage produces either validation or a precise revision target—not a vague “more work is needed.”

SECTION 14: CONCLUSION

14.1 What This Paper Establishes

Principia Recursionis v5.0 establishes the following:

Proved: The variational derivation of a master recursion from three resource constraints (bounded information, thermodynamic work, and model complexity). The well-typedness of the operator composition in finite-dimensional Hilbert space. The firm non-expansiveness of the proximal block. Forward invariance under explicit gain and drive bounds. Contraction with concrete parameter bounds for the linear-Gaussian instantiation. Consistency of the recursion with unitary quantum dynamics, classical Hamiltonian flow, active inference, soft RL, and replicator dynamics as special limiting cases.

Motivated but not proved: The interpretive mapping from abstract Lagrange multipliers to named cognitive constructs (Valuation, Desire, Drive, Meaning, Root Regulation). The complex projective structure prerequisite for the information-geometric derivation of unitarity. The global extension of KAM persistence beyond the local near-integrable regime.

Predicted and testable: The Drive–Progress Coupling Law (1>0 in Eq. 9)—that vigor increases with structural learning progress independent of reward rate and cost. The entropy-relief effect of Meaning (L>0)—that structurally compressible tasks reduce perceived effort at matched reward. The precision-modulation effect of Root Regulation—that improved allostatic state sharpens attention and lowers decision thresholds.

14.2 Where This Sits in the Scientific Landscape

This work occupies a precise position between Friston’s (2010) Free Energy Principle and Shenhav et al.’s (2013) Expected Value of Control framework. It inherits the free-energy architecture of the former but adds explicit typing of operators and novel kinetic (Drive) and salience (Desire) terms. It extends the latter’s decision-control framework to continuous-time, multi-operator dynamics with physiological grounding.

It does not supersede either framework. It proposes a more structured operator calculus that may subsume them as special cases—a hypothesis to be evaluated by whether the richer operator structure generates better fits to multi-modal behavioral and neural data than the simpler frameworks.

14.3 The Honest Scope

The claim is not to have derived a “law of becoming” or a “unified theory of reality.” Those are rhetorical formulations from earlier versions that overstated the case. The present claim is precise: this is a typed, variational, testable formalism for bounded adaptive agents that makes novel predictions, connects to established theories as limiting cases, and specifies conditions under which it is false.

That is sufficient to be good science.

Plain Language Summary — Section 14

What does this paper actually deliver? Mathematically: a well-defined, consistent update rule for adaptive agents, with proofs of stability and explicit conditions for contraction. Empirically: a specific, pre-registerable experiment that no existing theory fully predicts, offering a clean test case. Philosophically: a rigorous formalization of the intuition that meaning, drive, and self-regulation are not soft add-ons to cognition but structural components of any resource-bounded system that persists over time. What it doesn’t deliver: a complete derivation of physics from consciousness, a proof that “love is a mathematical necessity,” or a unified theory of everything. Those are interesting ideas worth exploring—but they are future work, not present results. The present results are hard enough and good enough to stand on their own.

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